How do you choose the right gain for root locus design?

1 What is root locus?

Root locus is a graphical method that shows the possible locations of the closed-loop poles of a system as the gain of a feedback controller changes from zero to infinity. The closed-loop poles are the solutions of the characteristic equation of the system, which depends on the open-loop transfer function and the feedback gain. The root locus plot helps you to analyze the stability, damping, and transient response of the system for different gain values.

2 How to draw root locus?

In order to draw the root locus plot, you must adhere to certain rules based on the properties of the characteristic equation and the open-loop transfer function. These rules state that the number of branches in the plot is equal to the number of poles in the open-loop transfer function, that the root locus starts at the poles and ends at the zeros or infinity, and that it is symmetrical about the real axis. Additionally, it crosses the imaginary axis at points where the real part of the characteristic equation is zero. The angle criterion states that when departing or arriving at a pole or zero, the sum of angles of vectors from poles and zeros to a point on the root locus should be an odd multiple of 180 degrees. Finally, breakaway or break-in points can be found by solving for zero in the derivative of the characteristic equation. You can either draw this plot by hand or use software like MATLAB or Python to generate it automatically.

3 How to choose the gain?

Once you have drawn the root locus plot, you can use it to choose the gain that meets your design specifications. For instance, if you want a certain damping ratio, natural frequency, settling time, or overshoot for the closed-loop system, you need to find the point on the root locus that corresponds to your desired performance and then calculate the gain that produces that point. Graphical or numerical methods like using a ruler or a protractor to measure the distance or the angle from the origin to the point on the root locus and then applying the magnitude or the angle criterion, using a trial-and-error approach to adjust the gain until you obtain the desired point on the root locus, or using a software tool like MATLAB or Python to plot the root locus with different gain values and then zoom in or out to find the desired point and the gain can be used for this purpose.

4 How to handle complex systems?

When faced with complex open-loop transfer functions or multiple feedback loops, drawing or interpreting a root locus plot can be difficult. To simplify the system before applying the root locus design method, you may need to utilize techniques such as partial fraction expansion to decompose a complex transfer function into simpler fractions, block diagram reduction to combine or eliminate blocks with common inputs or outputs, and feedback control design tools like MATLAB or Python to generate the root locus plot for complex systems. These tools also allow for interactive manipulation and modification.

5 How to verify your design?

To ensure that your root locus design meets your specifications and works as expected, you should plot the closed-loop transfer function to check its poles, zeros, and stability margins. Additionally, you should plot the step response to verify its rise time, settling time, overshoot, and steady-state error. Additionally, plotting the frequency response can help you check its bandwidth, gain margin, and phase margin. Finally, simulating your design with different inputs can help you observe its performance and robustness. Root locus design is a useful and versatile method for designing control systems that can achieve your desired performance. With the help of software tools like MATLAB or Python, you can draw, analyze, and modify the root locus plot.